Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $ x^{2} + 24 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 14 a + 23 + \left(9 a + 26\right)\cdot 29 + \left(14 a + 26\right)\cdot 29^{2} + \left(3 a + 12\right)\cdot 29^{3} + 29^{4} + \left(17 a + 1\right)\cdot 29^{5} + \left(5 a + 9\right)\cdot 29^{6} + \left(18 a + 15\right)\cdot 29^{7} +O\left(29^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 19 + 20\cdot 29 + 23\cdot 29^{2} + 7\cdot 29^{3} + 25\cdot 29^{4} + 10\cdot 29^{5} + 17\cdot 29^{6} + 9\cdot 29^{7} +O\left(29^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 12 a + 2 + \left(3 a + 11\right)\cdot 29 + \left(22 a + 22\right)\cdot 29^{2} + \left(12 a + 10\right)\cdot 29^{3} + 17 a\cdot 29^{4} + \left(25 a + 3\right)\cdot 29^{5} + \left(7 a + 16\right)\cdot 29^{6} + \left(23 a + 16\right)\cdot 29^{7} +O\left(29^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 12 a + 25 + \left(3 a + 12\right)\cdot 29 + \left(22 a + 15\right)\cdot 29^{2} + \left(12 a + 5\right)\cdot 29^{3} + \left(17 a + 12\right)\cdot 29^{4} + \left(25 a + 2\right)\cdot 29^{5} + \left(7 a + 28\right)\cdot 29^{6} + \left(23 a + 19\right)\cdot 29^{7} +O\left(29^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 15 a + 6 + \left(19 a + 2\right)\cdot 29 + \left(14 a + 2\right)\cdot 29^{2} + \left(25 a + 16\right)\cdot 29^{3} + \left(28 a + 27\right)\cdot 29^{4} + \left(11 a + 27\right)\cdot 29^{5} + \left(23 a + 19\right)\cdot 29^{6} + \left(10 a + 13\right)\cdot 29^{7} +O\left(29^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 10 + 8\cdot 29 + 5\cdot 29^{2} + 21\cdot 29^{3} + 3\cdot 29^{4} + 18\cdot 29^{5} + 11\cdot 29^{6} + 19\cdot 29^{7} +O\left(29^{ 8 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 17 a + 27 + \left(25 a + 17\right)\cdot 29 + \left(6 a + 6\right)\cdot 29^{2} + \left(16 a + 18\right)\cdot 29^{3} + \left(11 a + 28\right)\cdot 29^{4} + \left(3 a + 25\right)\cdot 29^{5} + \left(21 a + 12\right)\cdot 29^{6} + \left(5 a + 12\right)\cdot 29^{7} +O\left(29^{ 8 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 17 a + 4 + \left(25 a + 16\right)\cdot 29 + \left(6 a + 13\right)\cdot 29^{2} + \left(16 a + 23\right)\cdot 29^{3} + \left(11 a + 16\right)\cdot 29^{4} + \left(3 a + 26\right)\cdot 29^{5} + 21 a\cdot 29^{6} + \left(5 a + 9\right)\cdot 29^{7} +O\left(29^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(2,7,4)(3,8,6)$ |
| $(1,8,5,4)(2,3,6,7)$ |
| $(2,3)(4,8)(6,7)$ |
| $(1,7,5,3)(2,8,6,4)$ |
| $(1,5)(2,6)(3,7)(4,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,6)(3,7)(4,8)$ | $-2$ |
| $12$ | $2$ | $(2,3)(4,8)(6,7)$ | $0$ |
| $8$ | $3$ | $(1,6,4)(2,8,5)$ | $-1$ |
| $6$ | $4$ | $(1,8,5,4)(2,3,6,7)$ | $0$ |
| $8$ | $6$ | $(1,7,2,5,3,6)(4,8)$ | $1$ |
| $6$ | $8$ | $(1,7,4,6,5,3,8,2)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
| $6$ | $8$ | $(1,3,4,2,5,7,8,6)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
